发表机构
HSE University（高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该数论研究针对满足特定条件的不可约多项式，证明了区间$(X/2,X]$中存在足够多连续整数使多项式取值均为合数，改进了前人结果且适用于$f(t)=t^2+1$。
AI 中文摘要
设$f\in\mathbb Q[t]$为正次数、正首项系数且在所有整数处取整数值的不可约多项式。我们证明存在常数$c_f>0$，使得对所有足够大的实数$X$，区间$(X/2,X]$中至少包含$c_f\log X\log\log X$个连续整数$n$，满足所有$f(n)$均为合数。$\log\log X$的指数与$f$的次数无关，这改进了Ford和Gabdullin的先前结果——将他们的$\log\log X$的小固定幂次替换为完整因子$\log\log X$，该结果尤其适用于$f(t)=t^2+1$。
英文摘要
Let $f\in\mathbb Q[t]$ be an irreducible polynomial of positive degree with positive leading coefficient, taking integer values at every integer. We prove that there is a constant $c_f>0$ such that, for every sufficiently large real $X$, the interval $(X/2,X]$ contains at least $c_f\log X\log\log X$ consecutive integers $n$ for which all values $f(n)$ are composite. The exponent of $\log\log X$ is independent of the degree of $f$. This improves the previous result of Ford and Gabdullin by replacing their small fixed power of $\log\log X$ with the full factor $\log\log X$. In particular, the result applies to $f(t)=t^2+1$.
